Re: Box-counting dimension of random and Gaussian data points?

From: Ross Evans (rossYOUKNOWHATTODOevans_at_austin.rr.com)
Date: 12/09/04 

Date: Thu, 09 Dec 2004 02:45:13 GMT

My white-noise data points were actually one-dimensional, but not a time
series. The white noise points were downloaded from random.org, not created
with a PRNG. For the Gaussian set, I tried several methods: One method
simply added n of these white-noise points and divided by n for each
Gaussian point.. Another method used two of these uniformly random points
at a time and transformed them into Gaussian values using the Box-Muller
algorithm.

Either way, the resulting set of Gaussian values produced slopes in the
range of .92 when run through a box-counting program.

I really do not think my difficulty lies in any artifact of my Gaussian
sets. Rather, there is a fundamental difference between uniform white noise
and Gaussian brown noise. I understand how the former produces a
box-counting slope of 1. The latter does not, because the space is
inherently more sparse in some areas.

With the white noise, as the box size gets smaller, the slope starts at
exactly unity until the boxes are small and numerous enough to produce some
empty boxes. With the Gaussian noise with the same number of data points,
the empty boxes start to occur sooner, with larger box intervals. So the
box-counting slope is lower.

I have not yet tried a cloud of x,y points, but it seems obvious that I
would get a similar result as for a set of points in one dimension.

"Roger Bagula" <tftn@earthlink.net> wrote in message
news:41B72988.4060200@earthlink.net...
> Dear Ross Evans,
> Are you measuring a time series as:
> s[n]->{s[1],s[2], s[3],...,s[n]}
> Or
> {r[n],s[n]},-> {{r[1],s[1]},{r[2],s[2]},{r[3],s[3]},...,{r[n],s[n]}}
> The first will be one dimensional and the second should be two
dimensional.
> How are you producing your white noise/ normal variates?
> Different randomization method of producing your distribution will
> produce different
> results.
> Ross Evans wrote:
>
> >I need help understanding something.
> >
> >The box-counting dimension of a set of random data points -- random in
the
> >sense that they are white noise, equally likely to occur anywhere within
an
> >interval -- is theoretically 2. That is computed by adding 1 to the
slope
> >of the log/log plot of the single-variable data points, counted at
various
> >sizes of box measurent..
> >
> >That works out empirically for me. My measured slope of such a set of
data
> >points is very close to 1.
> >
> >But what about data points that are distributed not as such white noise,
but
> >as Gaussian-distributed data centered around a mean. These data points
> >visually do not fill the space as completely, because by definition they
> >tend to be clustered around the mean. And when I measure the slope of
such
> >a set of points, it is only about .92.
> >
> >But the literature generally seems to expect a box-counting dimension of
2
> >for Gaussian data, not just white noise. I don't get it.
> >
> >
> >
> >
>
> --
> Respectfully, Roger L. Bagula
> tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel:
619-5610814 :
> alternative email: rlbtftn@netscape.net
> URL : http://home.earthlink.net/~tftn
>
>
>

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